I agree. I would use one of the commercial moss killers, which are
effective. Raising the PH will also kill moss, but the level to kill
it quickly will leave the soil with a PH higher than desirable for
grasss.

It's also not true that the garden type lime products have no effect
on PH. They do, they just take a long time to work, which is why you
can apply them anytime without burning the lawn, which is not true of
contractor's lime.